
is time constant for a mixture traffic reflecting how quickly vehicles adjust their velocity to the desired value, which depends on both
time constant \(\tau _{\rm {acc}}\)  of ACC vehicles and time constant \(\tau _{\rm {m}}\)  of
manual vehicles. \(\tau _{\rm {mix}}\)  also depends on \(\alpha \) , the percentage of ACC vehicles to total vehicles. Fig. REF  shows the relations of these two parameters.
Parameter \(q_{\rm {in}} > 0\)  is a constant external inflow.
The traffic flow observes the mass conservation law [1]}, which is described by equation (REF ) and momentum equation derived from the velocity dynamics of ARZ model [2]}, which is described by equation ().
To stabilize the traffic flow so as to mitigate the stop-to-go wave in the congested regime, we will design a control input actuating on the ACC-equipped vehicle by manipulation of the time-gap of ACC-equipped vehicles from their leading car, denoted by \(h_{\rm {acc}}(x,t)> 0\) .
Input delays sometimes occur due to information transmission or driver's response to control instruction [3]}. Hence, we consider time delay \(D>0\)  in the model and design control which can compensate the input delay.
